Reduced-Rank Tensor-on-Tensor Regression and Tensor-variate Analysis of Variance

TL;DR

This paper introduces a tensor-based regression framework with low-rank structures and tensor-variate normal errors, enabling advanced analysis of multivariate tensor data and applications in neuroimaging and facial recognition.

Contribution

It extends classical multivariate regression to tensor data by imposing low-rank tensor formats and modeling errors with tensor-variate normal distribution, facilitating tensor-variate analysis of variance.

Findings

Identified brain regions linked to cognitive and emotional factors in fMRI data.

Distinguished facial characteristics related to ethnicity, age, and gender.

Developed efficient algorithms with known asymptotic properties.

Abstract

Fitting regression models with many multivariate responses and covariates can be challenging, but such responses and covariates sometimes have tensor-variate structure. We extend the classical multivariate regression model to exploit such structure in two ways: first, we impose four types of low-rank tensor formats on the regression coefficients. Second, we model the errors using the tensor-variate normal distribution that imposes a Kronecker separable format on the covariance matrix. We obtain maximum likelihood estimators via block-relaxation algorithms and derive their computational complexity and asymptotic distributions. Our regression framework enables us to formulate tensor-variate analysis of variance (TANOVA) methodology. This methodology, when applied in a one-way TANOVA layout, enables us to identify cerebral regions significantly associated with the interaction of suicide attempters or non-attemptor ideators and positive-, negative- or death-connoting words in a functional Magnetic Resonance Imaging study. Another application uses three-way TANOVA on the Labeled Faces in the Wild image dataset to distinguish facial characteristics related to ethnic origin, age group and gender.